Original ideas about the universe did not consider it as a mathematical model. Anaximander (610-546BC) was the first to propose a mechanical model for the world, envisaging the earth as the flat top surface to a cylindrical platform floating very still in the centre of the world. He introduced the concept of infinity to avoid Thales’ problem of what the earth may be floating in, although this paradigm was at first largely rejected. Anaximander did, however, contribute to the formation of other paradigms: being the first to realise the sun was a large mass, rather than a flat disc, and believing in spontaneous formation and dissipation to worlds, which moved eternally, “for without movement there can be no generation, no destruction”.

These anthropocentric worlds arose from the principles of self-extension into the ever-surrounding world. The same concept is evident upon early maps of the world, which were seldom more geographical than spiritual. Egypt, Arabia and Europe all founded early cartographic systems whereupon the country which they inhabited was central, the most accurately drawn and least adapted to geometric frivolity. Later, this anthropocentricism became so integrated into social acceptance that it founded evidence for divine logic and was thus difficult to displace without refuting religious fundamentals. The universe was naturally an extension of terrestrial cartography, and until Greek discussion in the fourth century BC macrodistances concerned only sailors, astronomers, philosophers and theologians. Hence it did not seem unreasonable in the sixth century BC to decide the celestial bodies be spherical on purely aesthetic principles. Whilst arguably this paradigm leap is nonsensical, the entire episode exemplifies the extent to which social and religious influences can dominate scientific direction, and do so not necessarily pejoratively.

Anaximander claimed his floating world to remain still “in the same place because of its indifference”, an important conjecture because it attributed human qualities to cosmological components, which is a further thought experiment using self extension. Social and religious factors were farther integrated into the thoughts of man during his early experimental developments. Aristotle (384-322BC) also believed in “developmental character” of natural motion – movement to fulfil one’s nature, not unlike early Chinese views of the world as a giant, interconnected life form, in which people, society, environment and science were constituent process of its overall functionality. But Aristotle was an anti-determinist, believing not only that people had free will, but that objects were governed by purpose and intent. This was an “animistic” perspective, attributing soul-like properties to objects and processes although frequently woolly, such as explaining the rose as red due to its property of redness. Later corpuscular theory, however, allows properties such as colour to be “the result of how our sensory apparatus actively processes impressions deriving from the real, primary realm”, the primary realm being definitive qualities such as size, shape and motion, whereby qualitative properties were subjective. Levy, meanwhile, argues in 1932 that to allot “properties” to objects is “unnecessarily abstruse”, since it attributes them ownerships which are non dependent upon observation and circumstances, but that properties should instead be studied as isolates of the wholes to which they belong. Aristotelian natural philosophy was a literally anthropocentric system, consisting of planetary crystal spheres upon which they revolved around a fixed, spherical world, for “only perfect circular motion was possible in the incorruptible superlunary sphere”. Spherical world conclusions were eventually established by navigational observation of sinking and rising of objects beyond the horizon, which revised the understanding of “down” and replaced the Anaximanderian cylinder. His eternal movement was also revised by Aristotelian cosmology, although giving rise to an inherently contradictory system.

Aquinas (1225-1274) wrote that “the Philosopher says things which are in a state of perfection possess their good without movement, but mechanistically, the incorruptible celestial bodies cannot be simultaneously still and in perpetual perfect spherical motion. This was not Aquinas’ concern, however: he was preoccupied with the pre-Christian leniency to Christian thoughts, and worried upon the difficulty in the Ascension of Christ through the impermeable heavens whilst “Christ was in a state of perfection” and thus devoid of motion. There were other religious barriers, such as how the world could have been made without introducing change to the unchanging superlunary sphere, which presented the question as to how the world could exist without being made by God. Its perfection in itself proved the existence of God, though it was its perfection which marred the theory. Ptolemy’s (83-168AD) introduction of hypothetical epicycles to the exactly spherical orbits around the earth partially rescued Aristotelian theory, for whilst the epicycles were mathematical models for planetary motion and not intended to be real representatives their impression also clashed with that of the crystal spheres.

Under these social pressures it would seem explicable to introduce immediate acceptance of Copernican theory, though his heliocentric model was in fact no less socially indulgent than the Aristotelian model. In view of this, it was primarily considered with much suspicion and accepted only on the premise as a calculatory aid. The rediscovered Ptolemaic system was complex, unwieldy and inaccurate, but Copernicus’ (1473-1543) model was in actuality no more accurate or less cumbersome than its predecessor. That it was accepted at all, even as a ridiculously contrived mathematical hypothesis, represents not its superiority in form, but the demand for the need it was designed to fulfil. Copernicus devised his model as a reaction to the social pressures for increasing the accuracy of calendars and sea navigation for exploration. He perhaps proposed the theory which had little mathematical prevalence due to its social functions of explaining the qualitative features of the “limited motion” of inner planets, retrograde motion in the form of links and kinks amongst the orbits of outer planets and the precession of the equinoxes. Even so, it retained several Ptolemaic features, including epicycles to correct circular motion. This Classical hangover arose from dependence upon the inherited paradigms and inability for Copernicus to free himself from what was essentially a form of divine arrogance. The principle of the relegation of the earth to a “mere planet” from a geo- to a heliocentric model upset and shocked a lot of its critics, who felt it was antireligious and that “the earth’s motion… violates the first dictate of common sense”, for “at the slightest jar of the earth, we would see cities and fortresses, towns and mountains thrown down”.

Essentially, heliocentricity was a compromise, offering to retain features such as “Special Earth functions” – the intersection of planetary orbits at the centre of that of the earth – at the expense of revolving the earth around a centre, which revolved around another centre, which orbited the sun. It’s complexities arose from an unwillingness to accept the model’s implications to the earth’s delegation, but its original repositioning necessary to in some way maintaining the astronomical discipline. Newtonian physics were later to refute this system of multiple revolutions, but in Copernicus’ day the motive for planetary motion which would eventually appear as the force known as gravity was still a debatable topic, allowing greater imaginative freedom. Notwithstanding this, Aristotelian animism still impacted sources of influence and explanation, and his sub- and superlunary separations forbade the comparison of micro- to macrosystems, such that the same laws of physics observable in the real world were never considered applicable.

The social and religious influences acting upon Copernicus do not explain specific changes that he made. It probably suffices to say that he was accidentally receptive to principles which later developed into paradigms due to a flailing search for adaptations to promote the branch of astronomy, which was considered at the time to be of a low prestige. The extent to which he was impacted by the world around him may only be sufficient to explain the existence of change rather than its future success or failure. Perhaps his methodology would have had a greater impact upon the choices he made in his model development and possibly improved its accuracy or led to further corruption of religious cosmological paradigms, for Copernicus in fact made fewer than one hundred observations himself, relying almost exclusively upon the data of others. This is a social privilege he possessed, but the inaccuracy of such data only allowed him to calculate a few new principles, including the tilted angle of the earth. But even this was rooted more in observational than mathematical proceeds and was established to account for the apparent motions of the fixed stars.

Although it is the Church who are traditionally scorned as suppressors of Copernican evolution, Catholicism had in fact been in the practise of taking non-literal readings from holy scriptures for a long time, which Copernicanism fully embraced. Osiander even inserted a preface to De Revolutionibus to the effect that it did not reflect reality, but was a calculatory aid. Nevertheless, in 1616 the book was forbidden to Catholic readership, and not removed from this list until 1835. The advent of Copernican theory was partially responsible for some of the cosmological conformity introduced into the Church for the first time between the fourteenth and sixteenth centuries. All of this sought to fulfil another social need – the need for religious authority. Sceptics may discuss the power-seeking role of the Church or the traditionalist constraints to which it adhered, but it is probably more functional to attribute the Catholic interference with the social strain which was a product of Protestantism. The Protestants held a much more literal reading of Biblical texts and were naturally more distressed by the propositions of Copernicus. Their splintering from the main Catholic religion was considered dangerous not only because of the competitive divide brought into the principal European denomination but also because the Protestants felt it expedient to expunge the inherent corruptions within Christianity, a threat to large sectors of particularly influential Catholics. Therefore it is hardly surprising that the Church sought to conciliate divisional tensions in their choice of what was really a political support, especially since there were few factors in favour of selecting either Copernican or Aristotelian cosmology over the other. The really questionable issue is what impelled them to become involved initially in astronomical politics, a discipline hitherto unconnected with religious particulars, except in that its pursuers would have been educated by religious faculties. Mathematics was a generally inferior study within social dynamics, so it is possible the Church felt it was an area over which they had full reign to use as a pacifying offering. Alternatively it’s novelty as a newly redeveloped scope may have captured their attention or the reservation contained within its cautious preface. For whatever principle, Copernicanism fell victim to the Chruch’s proceedings as a result of other dominating social influences. In fact it was not until 1992 that the Vatican finally recognised the validity of Copernican theory.

The acceptance of heliocentricity was undoubtedly accelerated by the interference of Galileo (1564-1643), the Copernican polymath, who invented the thermometer (1593) in addition to researching the pendulum, laws governing falling bodies, improving the hydrostatic balance for measuring specific gravity and conducting the first astronomical use of the telescope. Unlike Copernicus, whose De Revolutionibus was only published shortly before his death and had until then been withheld due to his uncertainly on its reception, Galileo operated in the public sphere, publishing a large number of works and achieving the title of Mathematician and Philosopher to the Grand Duke of Tuscany. The title of this position was elevated above that of the mere humble mathematician at his own request to be recognised for the wide spectrum of his achievements and in order to gain social status. In 1616, Galileo was ordered to abandon his support of heliocentricity, although he continued to investigate it in private study. At the 1624 ordaining of Urban VIII, Galileo obtained permission to publish a discussion on geo- versus heliocentricity arguments (Dialogue Concerning Two Chief World Systems, 1632), but it was concluded to be too partial: the book was banned and Galileo tried in Rome for heresy in 1633. The extent by then of his social influence was born out in that he was submitted to no torture, and only underwent life imprisoned by house arrest, where continued to publish a further text by smuggling it into Holland. Subsequently, Galileo is a useful study for weighing the extent of social influence on the evolution of mathematical models.

Whilst he operated by use of the public sphere, Galileo possessed the ability to dissociate himself and endogenous tendencies from the evidence by which he was surrounded. For example, he considered it “simply illegitimate to take the sun’s perfection as an undoubted premise for physical argument” and thus the religious and ceremonial ideologies of the past were extinguishable by the support of concrete findings.

The extent at which this dissociation broke down lay in his conservation of the belief in circular orbits which he claimed were required to “maintain the fabric of the cosmos in a state of perfection”. He attempted to expand and validate this paradigm, explaining the motion via circular inertial heliocentric orbits. But his telescopic use was the source of his contribution to Copernican theory, and his views fully explicable by his individual experimentation. The problem which Galileo addressed was that of introducing his own convictions into the social sphere, as it were, constructing paradigms from schema. Through his telescope, Galileo observed smaller, more distant stars, disproving the idea of a fixed background and reintroducing the “morally disorientating effects of the idea of infinite space”. He also discovered and mapped the moon’s topography and discovered sunspot imperfections revealing mutability (they came and went) and motion of the sun’s rotation. The addition of Jupiter’s moons and the phases of Venus, which, whilst explicable by other systems, appeared to support Copernicanism. Receivers of these new data sought comprehensively for alternative theories: perhaps they were “vapours raised from the earth and attracted the sun”? Galileo’s parallax obtained the spot-sun distance to be too small for the other favoured theory, of orbiting planets. This was a substantial blemish on Aristotelian theory: though it did not invalidate the geocentric system, it did invalidate the immaculate authority of its architect by questioning the distinction between the terrestrial and celestial systems of the sub and superlunary physics. Yet it’s loss allowed micromodelling upon earthly systems to help explain astronomical dynamics.

Unfortunately, Galileos’ alterations and additions intruded upon Neoplatonic beliefs by reducing the Copernican endowment of supremacy to the sun, which in this belief system symbolised the deity. Thus whilst simultaneously giving evidence for the theory, Galileo aggravated public opinion through his consecration of the Aristotelian religious implications, forcing the opinion that Copernicanism was “too strange and uncongenial for immediate acceptance”. The notable use of the word “immediate” is striking here because it suggests the gradual evolution of the World Problem into the heliocentric system. Society is, in fact, not set against the theory, merely unwilling to progress blindly into unexplored regions. It can hardly be viewed as injudicious for receivers of this reformation to be cautious and wary about its limitations. What is more, Galileo’s evidence could only be approved by trained astronomers, narrowing still the band of individuals who could contribute to the World Problem analyses. Spread of ideas was socially limited by the infrequency of printing, which though originating in 220AD in China, and appearing in Egypt in the 4th century and Arabia in the 9th only became established in Europe by the thirteenth century. What is more, the research group into astronomical behaviour was limited: no women or labourers, who were illiterate could play any part. Amongst those to whom text and education was available to allow them to form judgement most were biased by preconceived ideas wrought from their own studies and ideas.

Brahe (1546-1601) was one such man who disapproved of the Copernican system, creating his own Tychonic system to fulfil the social need for an intermediate compromise, expelling the crystal spheres, yet maintaining geocentricity. Brahe’s rejection of heliocentricity was founded upon reasonable arguments. He was unconvinced by the removal of astronomy from the theological realm, the lack of the stellar parallax from the extremes of the earth’s orbit, seemingly supporting a stationary status, and the vast waste of space Galileo implied lay between the stars and Saturn

However, his accumulation of data was hugely accurate and superior to any other, with accurate measurements for fixed stars within 1′ of an arc or better and planetary positions within 4′ of an arc – an amazing achievement for measurements taken with the naked eye. Brahe set the tradition for more accurate observational data which eventually cumulated in support for the Copernican model through such findings as the new star of 1572, which upset the immutability of the heavens (less careful observers claimed their parallax put it below the moon in the sublunary sphere!). He also created larger, stabler and better calibrated observational technology and improved reliability of data, correcting and expanding tables to free the ancient reliance on old data

Kepler (1571-1630) was a pupil of Brahe who, after his death, finished Brahe’s Rudolphine Astronomical Tables, the most accurate catalogues and calculatory devices of the time. He also introduced further adaptations to Copernican theory which depended heavily upon the accuracy of Brahe’s data. Kepler himself was a patient experimenter, founding the principle of mathematical rigour in scientific work. He was motivated by a sense of universal harmony which he felt fully explicable in mathematical justness. His Mysterium Cosmographicum, five platonic solids inside spheres to model the divine perfection of the orbit distances from the sun within 5% accuracy were a function and result of this harmony. Whilst this concept is clearly a model, and Kepler did not see the universe constructed of platonic solids, his operations upon heliocentricity make it doubtful whether they may still be considered as models, rather than reflections of astronomical realities.

In 1610, Kepler was given a telescope by Elector Ernest of Cologne, which he used to study its design, optics and use for astronomical research. Yet Kepler’s greatest achievements were not observational, but calculatory: he discovered the inconsistency of geocentricism within Copernicanism, abolishing Special Earth functions and finding whilst Copernicus measured the earth’s eccentricity from the sun, all the other planets were measured from the earth. By altering this to the sun, apparent variations disappeared, and Kepler located fluctuations in the compound circular orbits varying by a familiar mathematical model, leading him to the construction of elliptical orbits wherein the sun occupied one of the two foci – an effective rescue of the Copernican model. What is more, from this he deduced that equal areas were swept out by planets across increments of time, creating a mathematically dependable system which was now applicable to reflect reality. By Kepler’s postulate of the existence of magnetic forces between the planets and sun, driving them along their orbits or earth pushed round by an “anima motrix”, or sun ray, it is clear that the model had become a physical phenomenon, something believed actually reconcilable with earthly laws.

Kepler was a Lutheran, who experienced habitual religious persecution. The 1598 purge forced him to leave Graz for Prague; he returned a year later, but was forced to leave again. His uncannily accurate 1595 calendar was also considered with mystic suspicion and though he was made Imperial Mathematician in 1611 he also faced the stress of his mother’s accusation of witchcraft in 1615 until her exoneration in 1621. It is probably fair to suggest his social relations were poor and his work is largely independent of anything but inherited paradigms. As social and religious factors evolve, they become less and less consequential in World Problem affairs.

The function of these influences upon mathematical models was chiefly to retard the acceptance of Copernican theory, a factor which allowed it to be better certified before it was allowed to be elevated to a paradigmatic status, potentially beneficial in the construction of modern methods of scientific collection. It’s influence on the development of the theory does not extend to any great proportion and was mainly directed towards catalysing new conceptions, rather than in any specific way opening the way to heliocentricity. The social and religious factors acting upon earlier cosmologists were much more far-reaching in their integration into the isolate which later developed into the branch of astronomical physics. However, such impacts were distributed in a constant flux, so where Neoplatonism accepted Copernicanism to aid honour of the sun, it rejected the destruction of celestial purity by Galileo’s evidence and expansion of the theory. However, this reveals not so much an influence as a reaction to the various adaptations. Social and religious paradigms were influenced by mathematical models just so far as mathematical models were influenced by social and religious paradigms. The social rate of adaptation to new scientific paradigms is largely its limiting factor, though no more truly influential for mathematical models of the universe than any other alteration in human understanding. Technological limitations in executors, patrons and spread of information was hugely restrictive to model development into a realised reality. However, these factors raise the question as to whether technological influences are a social, economic or World Problem affair in themselves.

It is not unjust to suggest that religious maintenance was secondary to social influences upon science, although the degree of the integration of one into the other makes them largely indistinguishable. However, it is legitimate to claim the Church’s chief involvements occured principally upon social grounds rather than religious ones and that the impact of the religious condemnation of Galileo was more subtle in his day, whilst modern evolution of his views has allowed him to be considered martyred to their cause. For many historical characters involved in creating and adapting mathematical models of the universe, personal influences were equally so influential as social ones. Copernicus’ lack of experimentation founded the more specific format of his theory where social pressures only forwarded its presentation. However, it was the availability of older data which permitted this technique to be employed, a social influence born from the previous investigations of others. Yet arguably Copernicus would not have made measurements and thus deduced his theory without them, so it may be concluded that personal, technological and social influences were equally assertive in their influence upon mathematical models, wherein religious influences are a subset or coagulation of social principles.